Question

consider a closed rectangular box with a square base with side x and height y.
a. find an equation for the surface area of the rectangular box.
s(x,y)=

b. if the surface area of the rectangular box is 192 square feet, find \\(\frac{dy}{dx}\\) when x = 6 feet and y = 5 feet.
\\(\frac{dy}{dx}\\)=

Explanation:

Step1: Calculate surface - area formula

A closed rectangular box with a square base of side $x$ and height $y$ has two square faces of area $x^{2}$ each and four rectangular faces of area $xy$ each. So the surface - area formula is $S(x,y)=2x^{2}+4xy$.

Step2: Differentiate the surface - area equation with respect to $x$

Since $S = 192$ (constant), differentiating $2x^{2}+4xy=192$ with respect to $x$ using the sum rule and product rule. The derivative of $2x^{2}$ with respect to $x$ is $4x$, and for $4xy$ using the product rule $(uv)^\prime = u^\prime v+uv^\prime$ where $u = 4x$ and $v = y$, we get $4y + 4x\frac{dy}{dx}$. So, $4x+4y + 4x\frac{dy}{dx}=0$.

Step3: Solve for $\frac{dy}{dx}$

First, simplify the differentiated equation: $x + y+x\frac{dy}{dx}=0$. Then isolate $\frac{dy}{dx}$: $x\frac{dy}{dx}=-x - y$, so $\frac{dy}{dx}=\frac{-x - y}{x}$.

Step4: Substitute the given values of $x$ and $y$

Substitute $x = 6$ and $y = 5$ into the formula for $\frac{dy}{dx}$. $\frac{dy}{dx}=\frac{-6 - 5}{6}=-\frac{11}{6}$.

Answer:

a. $S(x,y)=2x^{2}+4xy$
b. $-\frac{11}{6}$